Question

A) Using Biot Savart’s law derive the expression for magnetic field in the vector form at a point on the axis of a circular current loop.
B) What does a toroid consist of ? Find out the expression for the magnetic field inside a toroid for the $N$ turns of the coil having the average radius $r$ and carrying a current $I$. Show that the magnetic field in the open space inside and exterior to the toroid is Zero.

Answer 1

Hint: A) Biot savart law is an equation, due to the current carrying segment the magnetic field will be produced. In this segment in vector quantity known as current element. By applying, Biot savart law the magnetic field produced at the point due to this small element can be known.
B) In toroid, we should find the expression for the magnetic field of $N$ turns of the coil. The toroid has a hollow circular ring in which a large number of turns $N$ of wire are closely wound. Toroid stores energy in the form of magnetic fields.

Complete step by step solution:
A) According to biot savart law, the magnitude of magnetic field at any point, due to current elements depends directly upon the current elements ($Idl$) the sine of angle is inversely proportional to the square of distance between current elements and point where the magnetic field to be calculated.
$dB \propto Idl$
$dB \propto \sin \theta $
$dB \propto \dfrac{1}{{{r^2}}}$
Combine all these above factor we get,
$ \Rightarrow $ $dB \propto \dfrac{{Idl\sin \theta }}{{{r^2}}}$
$ \Rightarrow $ $dB = k\dfrac{{Idl\sin \theta }}{{{r^2}}}$
$ \Rightarrow $ $dB = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{Idl\sin \theta }}{{{r^2}}}$
$ \Rightarrow $ $k = \dfrac{{{\mu _0}}}{{4\pi }}$
$ \therefore $ ${\mu _0} = {10^{ - 7}} \times 4\pi $
Where $k$ is the proportionality constant and ${\mu _0}$ is the permeability of a free space.
In vector form:
$d\vec B = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{Idl\sin \theta \hat r}}{{{r^2}}}$ where, $\hat r = \dfrac{{\vec r}}{{|r|}}$
$ \Rightarrow $ $dB = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{Idl\sin \theta \vec r}}{{{r^3}}}$
$ \therefore $ $dB = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{I(d\vec l \times \vec r)}}{{{r^3}}}$
Direct $d\vec B$ is perpendicular to plane containing $d\vec l$ and $\vec r$
Units and Dimension of ${\mu _0}$:
$dB = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{Idl\sin \theta }}{{{r^2}}}$
$ \Rightarrow $ $dB = \dfrac{{{\mu _0}Idl}}{{{r^2}}}$
$ \Rightarrow $ $\dfrac{{dB{r^2}}}{{Idl}} = {\mu _0}$
$ \Rightarrow $ $\dfrac{{{{\operatorname{Im} }^2}}}{{Am}}$$ = {\mu _0}$
$ \therefore $ $Tm{A^{ - 1}} = {\mu _0}$
In dimension:
$ \Rightarrow $ ${\mu _0} = $$\dfrac{{dB{r^2}}}{{Idl}}$
$ \Rightarrow $ $\dfrac{{\left[ {{M^1}{L^0}{T^{ - 2}}{A^{ - 1}}} \right]\left[ {{L^2}} \right]}}{{\left[ {{A^1}} \right]\left[ L \right]}}$
$ \therefore $ ${M^1}{L^1}{T^{ - 2}}{A^{ - 2}}^{}$
Hence this the Biot savart law for magnetic fields in the vector form at a point on the axis of a circular current loop.

B) In toroid the direction of the magnetic field inside is clockwise as a right hand thumb rule. The magnetic field should be tangent to then and magnitude is constant.
Let $B$ be the magnetic field inside the toroid,
By Ampere’s Law,
$\oint {\vec B.d\vec I = {\mu _0}I} $ Or
$BL = {\mu _0}NI$
Where,
$L$ is the length of the loop
$B$ is tangent
$N$ is number of turns
$I$ is current in the loop
Let,
$L = 2\pi r$
By applying $L$ ,
$B(2\pi r) = {\mu _0}NI$
Therefore $B = \dfrac{{{\mu _0}NI}}{{2\pi r}}$
In inside the toroid, the open space is encloses so, no current in the loop
Thus $I = 0$
Hence the $B$ is also became Zero.
That is $B = 0$
In exterior the toroid is in open space as it in each turn the current carrying wire and it is cut into twice by the loop.
Thus, in plane the current coming out is cancelled by current going into.
So, $I = 0$
$B = 0$
Hence, that the magnetic field in the open space inside and exterior to the toroid is Zero.

Note: A) In Biot-savart law, the magnetic field is generated at constant current it will relate the magnetic field to magnitude, length of an electric current. Biot-savart law allows both Ampere circuit law and Gauss’s theorem. It is also used to find the magnetic field intensity in a near current carrying conductor.
B) In toroid the field B is constant in magnitude and the ideal toroid is closely wound in turns then the $B = {\mu _0}nI$. Toroid coils work currently which have low frequency. Toroid works as an indicator when the frequency level is boosted. Toroid can store energy in the form of magnetic fields.